Monday, 9:00 – 12:00

Session 1 : What is a numerical table, from mathematical and linguistic points of view, and in the modern context of computers ?

Chair : Renate Tobies

Renate TobiesGeneral introduction to the mini-workshop

Dominique TournèsWhat is a numerical table ? Milestones for a historical research project

Abstract. In this introductory talk, I’ll present an assessment of the HTN project after 18 months and I’ll try to sketch some ideas for the continuation. What definition to adopt for a numerical table ? What kinds of tables did we encounter ? What methodological questions emerged ? What new book could we write to complete and to renew The History of Mathematical Tables (Oxford, 2003) ?

Karine ChemlaReflections on texts of tables based on Chinese manuscripts from early imperial China

Abstract. On the basis of ancient Chinese texts, I shall introduce various criteria that can be used for the study of the text of a table. I shall show that there were forms of text shaped to write down tables in ancient China, but that these texts differ from our expectations about such texts.

Liesbeth De MolWhat is a mathematical table in era of digital computing ?

Abstract. Is it true that, because of the speed in computing made available through the digital modern computer, there was and is no longer a need for mathematical tables ? It is clear that those involved with table making and/or digital computing during the pioneering years of the modern computer, were rather pessimistic about the future of projects such as the Mathematical Tables Project and mathematical tables in general, because of the possibility of high-speed computing. As Ida Rhodes recounts in an interview : "Do you know, [Lowan] said, what they do ? They don’t look up Tables. They actually compute each value ab ovo. And to me that sounded so impossible, so incredible [...]. To compute each value ab ovo. Not to have to look up one of our marvelous Tables. That sounded like [the] death knell." This pessimism about the future of mathematical tables, given the computer, is confirmed in the relatively recent volume on The History of Mathematical Tables, where the possible usages for such tables are reduced to that of the table-as-spreadsheet. But is this really all there is to say about mathematical tables in the digital era ? As I have argued before, the role of mathematical tables extends well beyond that of the table-as-spreadsheet, provided that one considers the possibility that there have been important changes in the way mathematical tables are computed, represented and used since the rise of the computer and that, as a consequence, both the (printed) table-as-calculated-aid and the table-as-data-presentation before the rise of the computer has evolved into another kind of table-as-calculated aid and table-as-data-presentation. But what exactly has changed ? What is a mathematical table in the digital era ? Does one need a fundamental change in the definition proposed in the ANR project Histoire des Tables Numériques, where mathematical tables are defined as : "tout type de texte établissant des correspondances entre deux ou plusieurs phénomènes qualitatifs ou quantitatifs, et s’attachant à disposer sur une surface plane, d’une manière donnée, des séries de valeurs numériques associées à ces phénomènes." The aim of this talk is to tackle this kind of questions. The approach here is not to consider the discontinuity : tables before and after the computer but rather the evolution : tables during the digital era. More specifically, I will focus on (the use of) "mathematical tables" during the digital era within mechanized mathematics, connecting some of the fundamental changes occurring within mathematics due to what I have called internalization and time and space squeezing on other occasions, to changes within mathematical tables – their methods of construction, their role, their applications and their representations.

Monday, 15:00 – 18:00

Chair : Thomas Sonar

Matthieu HussonNumerical tables and interpolation

Abstract. I studied two sets of double entry tables : one by Lignieris on the equation of planets ; the other by Muris on the calculation of real conjunction of Sun and Moon. Given the similar mathematical situation and the proximity of the two authors we would expect that they use the same interpolation procedure in the canons of their tables. Although both of them use difference number (some time "second order" difference of difference) and what I’ve called "oriented number" (to put it short some sort of positive and negative number) they do not present the same interpolation algorithms. It seems that the astronomical content of the numerical tables influence the way interpolation is conceived. Moreover the different manuscript copies of the same work do not necessarily present the same interpolation procedure (the more complex procedures are found in the earliest copies) pointing at a sort of quiet autonomisation of the subject of interpolation between the time of the conception of the work (early 14th century) and the time of the most recent copies (15th century). This example shows that interpolation is a specific topic which can help in the study of more general questions :

Interpolation techniques are used both in the making of tables and in the use of tables. The study of both (maker’s and user’s) set of techniques can help to address precisely sociological questions we have around tables. (For example if in some period and place the two set of techniques appears to be identical we have good ground to conclude that user’s and maker’s had the same sort of background mathematical education, if the user’s method are much more simple than the maker’s we may think that the user’s community is significantly wider or distinct from the maker’s community).

Interpolation techniques are influenced by the presentation of the tables (the problem of interpolation in a double entry table is obviously different from the problem of interpolating in a simple entry one) and reciprocally interpolation techniques influenced the setting of tables (for example "difference number" can be added to speed up the interpolation procedure, specific tables can be conceived). Thus semiotic questions we have around tables can also be addressed through the interpolation lens.

There are of course many mathematical questions around theses type of techniques. Interpolation can be a way to study the relation of discrete and continuous mathematics. Interpolation techniques are also part of the story of the function concept.

Abstract. Al-Samaw’al, a 12th-century Iranian scientist best known for his contributions to algebra, wrote an inflammatory treatise near the end of his life entitled Exposure of the Errors of the Astronomers in which he criticized virtually every one of his astronomical contemporaries and predecessors. Among his complaints was the use of approximation methods in the construction of sine tables. To avoid this problem, Samaw’al redefined the circle to consist of 480 parts rather than 360°. However, close examination of the entries in his table reveals that he generated them simply by interpolating the entries of a conventional 360° table. Now, numerical tables are unusual artifacts in the history of science, since much of the activity that generated them is undocumented in the primary literature. This raises the question : what can we really know about the ethical standards to which Samaw’al would have been subjected by his peers ? Is it fair to accuse him of academic fraud ?

Joachim FischerA Closer Look on the Computation of Napier’s Table of Logarithms

Abstract. John Napier (1550-1617) published his Mirifici canonis logarithmorum descriptio in 1614 ; this book contains a rather short introduction, several applications and, of course, Napier’s table of (his) logarithms for the 5401 sines of angles a, a = 0°0’ (1’) 90°0’. Posthumously, his Mirifici canonis logarithmorum constructio was published in 1619 ; here the process of the computation of his logarithms was described in more detail, albeit in strictly Euclidean style. Although almost 400 years have passed since, there is still something left for us to learn and understand. Several questions have been treated in the course of time : Why do Napier’s logarithms deviate (partially) from the correct figures ? Which table of sines did Napier probably use ? What if he had not committed a few minor mistakes during calculation (with very different consequences, though) ? And so on. But a few points have never been considered in detail. Why and where do the sines printed in Napier’s table differ from the ones probably used in his calculations (and there are differences) ? Why does Napier’s method of approximating a hitherto completely unknown function within a prescribed range of accuracy really work ? How do his logarithms depend on the accuracy of the sines used ? Which one of the two methods, as described in the Constructio, for extending the table from a starting interval (chosen a priori) to the rest of the table was actually used by Napier ? What might be the reason in, or the use of, recalculating Napier’s logarithms with higher precision (as done in the 19th century) ? Current investigations by my Ph.D. student Bärbel Ruess, based on previous work (Fischer 1997, 1998), could perhaps provide some answers…

Abstract. As early as 1820, at Cambridge, John F. W. Herschel (1791-1871) produced a volume of examples on the finite differences method. The analysis of this text - which seems at first to be tightly fostered by Laplace - and the way Charles Babbage (1791-1871) used it in the 1820s in constructing his first "difference engine", will help us to grasp the relationships between differential calculus and the calculus of finite differences at the time. The transition from the "difference engine" to the "analytical engine" will show what kinds of extensions were then made possible. The result of this analysis would improve our understanding of the algorithmical way of thinking what a "function" was, as this mathematical concept was so strongly present in the English algebraists works during the first part of the 19th century.

Tuesday, 9:00 – 12:00

Session 3 : Arithmetical tables and other numerical tables

Chair : Clemency Montelle

Micah RossSurvey of graphical and numerical tables in Egypt

Abstract. According to the definitions of informatics, tables order information by rows and columns. One implication of this definition is that information may be generated by transforming the elements of a row or column. Some Egyptian texts may conform to this model but insufficient background has been established to assume an understanding of Egyptian tabular compositions. Truly tabular texts are somewhat uncommon in Egypt and the tabular disposition of information remains one of the less studied aspects of Egyptian texts. Perhaps because of this relative infrequency, modern databases have indexed Egyptian texts by various fields (e.g. writing ground, date, provenance), but numerical and tabular texts have not merited special consideration. As an initial step to understanding the Egyptian standards for the construction of tabular texts, a definition of the genre of graphical or numeric tables must be established and the extant graphical and numerical tables must be surveyed. Particularly difficult are "table-like enumerations" which may be implicitly indexed. In these cases, the interpretation of the text may vary by the understanding of the index. A few previous studies have focused on the fractional tables of famous mathematical papyri and derived rules of composition from these examples. However, several unpublished texts challenge these rules. Finally, some compositions conform to strict definition of a numeric table but risk overwhelming the genre.

Christine ProustIs Plimpton 322 a table ?

Abstract. Various interpretations of the famous tablet "Plimpton 322", dealing with Pythagorean triples, were published last years. These interpretations are heavily based on the presupposition that the text is a table providing data for purposes which vary according to the different authors (theory of numbers, or school exercises, or others). I’ll offer a new reading of this text, based on the idea that it is a problem text with a statement and a complete solution. This interpretation results from an article by Britton, Shnider and myself (to appear).

Agathe KellerComposing material for the HTN book after the Imera International Workshop on Numerical Tables in Sanskrit

Abstract. This presentation will consider the work accomplished mid-December in a Marseille workshop from the perspective of the book HTN aims at publishing. Three main strands were highlighted in the workshop : South Indian traditions of numerical tables, astronomical table-texts, and a long pan-Indian evolution of sine difference tables. Much had to be left out, notably, magic squares and ephemeral computational tables. More generally, the interactions, complementations and differentiations between different kinds of numerical tables in Sanskrit sources will be raised. Consequently, how Sanskrit and more largely "Indian" tables should be described and thought of within chapters of the HTN book will be discussed.

Maarten BullynckTables in the theory of numbers : a research report

Abstract. I will present an overview of the histories of some specific numerical tables used in number theory, mainly from the 16th-17th century until today. Using existing case studies on factor and prime tables, tables of decimal periods and exponents of 2, as well as a work in progress on tables of quadratic forms, I will try to get at some more general questions, such as :

What are the relationships between the tables and the corpus of theory and techniques used in number theory ? Is it a static one (collection of materials vs. body of theorems) or a dynamic one (exploration and problem solving) ?

Can the observations about number theoretical tables be compared with or extended to other mathematical disciplines ? Or is there a specificity to number theoretical tables ?

Tuesday, 15:00 – 18:00

Session 4 : Ancient astronomical tables

Chair : Glen Van Brummelen

Matthieu OssendrijverBabylonian astronomical tables

Abstract. Cuneiform tablets from the Late Babylonian period (450-50 BC) contain the first written evidence of mathematical astronomy in the ancient world, in the form of astronomical tables with computed data for the Moon and the planets, and procedure texts with the corresponding instructions. In this presentation I explore the organisation and representation of data in the astronomical tables, the use of labels and other descriptive information in the tables, and the relations between the different functionally defined types of tables.

Nathan SidoliDiagrams and Tables in Ptolemy’s Almagest

Abstract. In this talk, I will address the question of how Ptolemy uses the combined tools of diagrams and tables to model the motion of the planets. In particular, I will address the questions of how Ptolemy could be convinced that the tables were an actuate representation of the model, and how we should understand the relationship between the entries in the table and the components of the geometric model.

Abstract. One important strand in the study of numerical tables in Sanskrit sources comprises of the so-called kosthakas or saranis, astronomical tables that rose in popularity from the twelfth century onwards. It has been argued that the prominence of these types of tables in Sanskrit astronomy was linked to Islamic inspiration particularly through the influence of the zij compositions. Unlike the standard Sanskrit astronomical formats that contained enumerations of important parameters and fundamental algorithms composed in verse, these works used spatial arrangement, ruled rows and columns, alignment, and accompanying explanatory prose to present precomputed data intended for practical astronomical application. We will survey this strand and examine some of the characteristics of these works through specific examples.

Mahesh KoolakkodluVakyakarana : An astronomical text with tables

Abstract. Indian astronomical texts are classified into three types : Siddhanta, Tantra and Karana. Numerical tables are found in most of the texts of these kinds. Among these, Karana (lit. "instrument") texts present tabulated values as a tool for making the computational procedure easier. These numbers are presented either in the form of verses or phrases. The latter method was commonly used in South Indian astronomical texts in tabulating the values which are useful for computing longitudes of the planets, aspects of eclipses, and so forth. One such text that uses this specific method of listing numbers is Vakyakarana. "Vakyakarana", as the name reveals, is a tool/manual with vakyas (phrases). The method prescribed in this text was widely used for making almanacs in south India. This anonymous text has a commentary of Sundararaja (c. 1300 CE). The commentary puts light on the method of construction and the use of alphanumerically encoded tables.

Wednesday, 9:00 – 12:00

Session 5 : Applications of tables in social and economic life (navigation, artillery, engineering)

Chair : Joachim Fischer

Thomas SonarOn the development of declination tables in early modern England

Abstract. In the beginning of the 16th century the only available navigational tables in England were so-called "rutters" which originated in France. These rutters contained information about the waters in the English channel and the Biscaya only. Taken up in England the rutters were enlarged by data from English coasts but it took until 1561 to comprise an English translation of Martin Cortes’ Spanish book Arte de navigar. Since the look-up of the sun’s declination in Cortes’ tables took three steps and due to the inexperience of English mariners the book was too advanced. Soon William Bourne comprised declination tables in his An Almanac and Prognostication, first published in 1567, in which only one table look-up was needed. Already in 1594 finding the declination of the sun in order to find ones latitude on sea was so commonplace, that we find a simple declination table in Thomas Blundeville’s His Exercises, containing six Treatises. The next large step forward was taken by Thomas Harriot who computed declination tables (called Regiment of the sun) from independent data which he himself took from astronomical observations. These tables were certainly the best in Europe but did not widely circulate. It is not known for sure but highly likely that Edward Wright knew Harriot’s tables. In his Certain Errors in Navigations we find fine tables of the sun’s declination which bear some characteristics of Harriot’s and which comprised the state of the art then achieved.

David AubinRange tables at the beginning of the 20th century

Abstract. In this talk, I would like to explore the production and use of ballistic tables in the first decades of the 20th century. At the end of the 19th century, I will argue, the science of ballistics was well established with standard textbooks and recognized international leaders. Besides theories and computing procedures, it also produced range tables whose utility for combatant troops will be examined. When World War I broke out, range tables proved inadequate to the task. I will examine why this was the case and discuss the measures taken to remedy this ballistics failure, including the appeal to civilian mathematicians.

Renate TobiesDevelopment and using of graphical and numerical tables in interaction between university and industrial research in the beginning of the 20th century

Abstract. Numerical and graphical tables became a generic component in the beginning of modern numerical mathematics. It has to underline that numerical and graphical tables were forerunner instruments if we compare their function with the modern computer. Such tables were developed to solve differential equations. They have been developed, used, and taught in interdisciplinary research seminars of applied mathematics at the University of Göttingen, when Carl Runge (1856-1927) became the first professor of applied mathematics at a German university in 1904. Collatz (1990) called Carl Runge "the founder of modern numerics". Graphical and numerical tables were used for solving problems in electrical engineering, geophysics, aerodynamics, physical chemistry, statistics etc. It was in this context that mathematicians first began to discuss, in concrete terms, what was meant by "applied mathematics". When, in 1907, practitioners of applied mathematics gathered at the University of Göttingen, they issued the following formulation : "The essence of applied mathematics lies in the development of methods that will lead to the numerical and graphical solution of mathematical problems". At the same time, tables as generic components of mathematical modelling flowed into the industrial research and were spread internationally. The lecture will explain basic trends and a survey of developed tables and the first published books in this field.

Denis BayartUsing Numerical Tables as a Medium in Client-Provider Negotiations in 20th Century Industries : the Case of Statistical Quality Control

Abstract. My intention is to go further into the analysis of numerical tables in the domain of Statistical Quality Control, which I have presented in Budapest (2009). I am interested in the practice which goes along with the tables, according to published accounts. In industrial quality control, numerical tables are used to construct a frame for the client-provider relations. The general procedure is called "acceptance sampling". When the provider delivers a quantity of goods (called "the lot") to the client, a pre-determined number of pieces ("the sample") is taken at random. Each element of the sample is examined with respect to pre-defined quality criteria and judged "good" or "defective". According to whether the number of defectives in the sample is smaller or greater than a pre-determined "acceptance number", the whole lot is accepted or rejected. The tables give the sample size and acceptance number corresponding to various lot sizes and probabilistic evaluations of the risk of making an inadequate decision. Starting from this simple scheme, many variations in the methods can be observed. Today SQC is an applied branch of Statistical Decision Theory, but it has developed original methods and cognitive tools in the course of its history, particularly when it has to be put to use "under commercial conditions" by agents who are not trained in mathematics or statistics. In an industrial context, uses and practice of the tables are not self-evident for actors under economic and time pressure, members of organisations which impose their own rationale. The design of the tables is well documented in numerous papers published in scientific or technical journals all along the 20th century. Mathematically speaking, working-out the tables does not seem to raise heavy problems. But the designers try to take into account in anticipation some features of practice which have been mentioned by users. This interaction produces many variations in the methods and cognitive tools according to practice situations. We have thus an opportunity to analyze the dimensions according to which the differentiation in the tables occurs. My documentation concerns situations where the tables are a structuring element in the client-provider negotiation. They are part of the administrative and legal frame. The economic dimension is also important, especially concerning long-term relations and large batch production. Then the selection of the sampling plan parameters is an important stake. The documented discussion around such stakes reveals a great variety in the interpretations of the numerical tables. My aim is to analyze this diversity, connecting it with the elements of the tables which appear to support it in practice.